Time-release Cryptography from Minimal Circuit Assumptions

Abstract

Time-release cryptography requires problems that take a long time to solve and take just as long even with significant computational resources. While time-release cryptography originated with the seminal paper of Rivest, Shamir and Wagner (\u2796), it has gained special visibility recently due to new time-release primitives, like verifiable delay functions (VDFs) and sequential proofs of work, and their novel blockchain applications. In spite of this recent progress, security definitions remain inconsistent and fragile, and foundational treatment of these primitives is scarce. Relationships between the various time-release primitives are elusive, with few connections to standard cryptographic assumptions. We systematically address these drawbacks. We define formal notions of sequential functions, the building blocks of time-release cryptography. The new definitions are robust against change of machine models, making them more amenable to complexity theoretic treatment. We demonstrate the equivalence of various types of sequential functions under standard cryptographic assumptions. The time-release primitives in the literature (such as those defined by Bitansky et al. (ITCS \u2716)) imply that these primitives exist, as well as the converse. However, showing that a given construction is a sequential function is a hard circuit lower bound problem. To our knowledge, no results show that standard cryptographic assumptions imply any sequentiality. For example, repeated squaring over RSA groups is assumed to be sequential, but nothing connects this conjecture to standard hardness assumptions. To circumvent this, we construct a function that we prove is sequential if there exists any sequential function, without needing any specific knowledge of this hypothetical function. Our techniques use universal circuits and fully homomorphic encryption and generalize some of the elegant techniques of the recent work on lattice NIZKs (Canetti et al., STOC \u2719). Using our reductions and sequential function constructions, we build VDFs and sequential proofs of work from fully homomorphic encryption, incremental verifiable computation, and the existence of a sequential function. Though our constructions are theoretical in nature and not competitive with existing techniques, they are built from much weaker assumptions than known constructions